Computer Variations is my first acknowledged computer piece, which I wrote during the same year that I began teaching at Queens College. The piece is a traditional set of variations, beginning with a theme that is transformed, and sometimes literally repeated, in each variation. Following the theme, there are seven variations.
At the time that I wrote it, the computer synthesis techniques available were quite limited, and the processes that I used in each variation are remarkably simple, although their implementation is very accurate and unsparing. All of the tones are generated by simple oscillators, the only timbre variation created by mixing waveforms with different harmonic partials. The theme is a three-voice tune lasting only 34 seconds. The ensuing variations make use of different envelope shapes, timbre changes, reverberation, and amplitude and frequency modulation to produce sharply defined sounds. All sounds are located in various places between the two speakers, sometimes traveling from one to the other. Reverberation is sometimes used to create the sensation of traveling into the distance, but I don’t think this process worked very effectively; nevertheless, the reverberated sounds are clearly differentiated from the others. The fifth variation consists of four-note chords that fade in and make asynchronous glissandos to the note in the next chord. The sixth uses four instruments that each have very different envelopes, ranging from half a sine wave to shapes that are mostly decay to mostly rise, with all notes also having other distinguishing qualities, such as amplitude and frequency modulation, and very exacting rhythms. The last variation is similar to the fifth without the glissandos: each note simply fades in and sustains for a different duration, so that the effect is a changing mosaic of chords.
In resynthesizing this work from the old computer outputs I found in my basement in the summer of 2004, I can recall the day-to-day problems that plagued my work when this was done. Three different computer synthesis languages were used, ranging from Music 4B, a similar program I wrote for the IBM 7040, and Music7, which I wrote for the XDS Sigma-7 computer at Queens College. While all these procedures could be translated into csound, I had a hard time remembering what all the different statements meant. The only storage medium that large data files could be saved on were magnetic tapes, and I had to travel to both Princeton University and Bell Telephone Laboratories to convert these tapes to sound. I then had to splice the magnetic tape segments of no more than two minutes each into the final result, which is 9 and a half minutes long. What a difference 37 years makes!
Macro Structure was written in 1971 but never adequately realized, since I had envisioned it for a four-channel playback system. At that time, it was synthesized and performed in stereo.
The basic idea of a "macro structure" is that a single line in a computer program specifies an entire series of operations, usually with different data that are expanded in the operations. In this composition, there are three basic ways in which this idea is realized: (1) Each "note" specifies a cluster of pitches transposed to begin from the indicated pitch. Different sections change from trichords to pentachords and to hexachords. (2) Each cluster has an envelope which allows each of the components and the totality to have prominence over a portion of the note. (3) The developing process continues through the entire piece, in a palindromic structure. The pitch of each component begins steadily, then starts vibrato. As the piece continues, both the amount and the speed increase, then decrease after reaching the climax in the middle.
Newer computers have allowed for synthesis in four discrete channels, and the work has been resynthesized. My original conception has now be realized, although it can only be heard in a concert with four channel playback facilities. In order to fully appreciate it, you will also need to sit as near to the middle of the playback venue as possible.
Freeze uses sounds created by filtering a pulse-like wave with variable filters that slowly sweep back and forth between two harmonics, constantly producing a change of timbre. Tones are often modified by other processes: amplitude modulation, travelling back and forth between the two speakers, and reverberation. The speed of variation of each property slowly changes over the course of the tones. The piece is in four sections, building to a climax at the end of the third section. In the first three sections, the speed and dynamics of individual tones increase by a factor of one-third over the course of the duration, which exactly parallels the amount of change over the entire section. In the fourth section, these properties decrease, so that the overall structure is a large crescendo or increase followed by a smaller diminuendo or decrease. The speeds of variation are different within each octave, with higher octaves faster than lower ones. There is a constant ratio of 1 to 5 between the speeds in each octave. This produces a distinct character for the tones within each octave. In the middle two sections, which are so similar that they appear more to be one large section, sequences are introduced, and the articulative properties are applied not to the individual tones within the sequence but to the whole, so that timbres change, for example, at a different speed from the attacking of new tones in the sequence, and both speeds change. The title "Freeze" is intended to reflect the redundancy of materials on which the piece is based, as if these are "suspended" or frozen while what changes is our perspective of them.
Timbre Study No. 3 is the third movement of Three Studies in Timbre, composed between 1970 and 1973.
Timbre is the overtone structure of a sound. Each of my studies in timbre is based on some different conception of musical timbre, particularly conceptions that can be explored more successfully in computer music synthesis than with other methods of sound generation.
Timbre Study No. 3 is based entirely on the use of harmonic partials that fade in and out in different ways over the course of each tone. Only the first twelve partials are used. One group of instruments continually attacks tones at a speed which is deliberately out of synchronization with the tempo. Another group gives tones a strong attack followed by an exponential decay, while a third group produces a crescendo followed by a diminuendo. The last and most important group plays long tones in which individual overtones cyclically fade in and out, producing a melodic interaction with the other music. Variations in timbre are coordinated to location changes and amplitude variations.
Timbre Study No. 3 is recorded on Opus One No. 47.
Canons is a four-movement piece written in 1974, shortly after I encountered frequency modulation synthesis. In this manner, simple timbral variations can be produced easily, and they are used extensively in the piece.
FM synthesis had been invented a few years earlier by John Chowning at Stanford University. Chowning’s outstanding contributions to computer music are among the most important in history. Long before Yamaha had adapted it to their first venture in electronic music synthesizers and even before his article on the subject was published in Computer Music Journal , an earlier version of his article was circulating among members of the computer music community. This was where I learned it, and this work was the piece in which I carried out all my initial experiments. Because the work was so long, I was unable to record or present the full work, but the fourth movement was recorded on Opus One No. 47.
In Canons 4, there are two independent sub-parts within each part. In addition, there are two modes of articulation for tones in each sub-part, providing the basic contrast of sounds heard in the piece: short tones with percussive attacks which fade away quickly, and long tones which increase and decrease both in amplitude and in beating. The overall harmonic effect is one of continuous change through complex chords in which each tone fades in and out at a different time and at a different speed. Rhythmically, the piece starts at a moderate tempo and progresses to a constantly slower unfoldment of materials, increasing the density while maintaining the same overall rate of change.
Improvisation on the Overtone Series is based on the idea of employing the overtones of fundamental frequencies in a manner analogous to the use of independent pitches. When beginning the work, I made two assumptions pertaining to the entire piece: (1) Overtones should be audible as separate tones, yet still contribute to the "color" of the fundamental frequency, which would last over a longer duration. (2) The order in which the overtones occur would be determined by the "harmony" of the passage in which the fundamental occurs. Only harmonic partials are used.
Within this framework, I designed two kinds of computer "instruments". The first "brings out" the partials one at a time by means of an amplitude control function. The shape of this function is a rise over the first 3/16ths of a cycle, a sustain for the same duration and a decay for the rest of the cycle. The "cycle" is the duration over which all partials enter and leave the tone in this fashion. Each tone in the piece usually goes through several such cycles over the course of its complete duration.
The second instrument attacks each partial (i.e. a sine tone) separately and sustains it for 3/8ths of the duration of a cycle; after this duration, the tone dies away completely, so that there is never the impression of a "floating background" as with the first instrument. In this case the impression of the fundamental does not emerge until a few tones have sounded.
Each of these instruments normally employs 16 partials, stretching from the fundamental up over a range of five octaves. In the last section of the piece, there is a passage where only partials 8 through 16 are used, on an instrument otherwise identical to the first type described above. The fundamental emerges clearly, but the timbre is more nasal in character.
All "harmonies" (simultaneous sounds) are derived from 3-note and 4-note chords. Sometimes more complex chords are formed by combining these, but in that case the overtone pattern is a combination of the two chords forming it. The overtone patterns I selected were chosen by placing overtones that create the "harmony" in question at the beginning of the cycle, so that they are heard as a separate "chord" but still blend in with the fundamental as part of the timbre of the sound. Following these tones, which always occur in descending order, the other overtones occur in a manner that tends to be dispersed over the entire five-octave range of the first 16 partials.
The piece is in three sections, which might be heard as independent "movements" of the entire work, although I intend for all sections to be heard together. The piece was synthesized in 1977 using the XDS Sigma-7 and IBM 370/168 computers at Queens College and the City University of New York.
Improvisation on the Overtone Series is recorded on Opus One No. 53.
The basic idea for Luminescence is revealed by the opening sound, an F above middle C, which is modified by a filter that moves from a point above the second harmonic to about the sixth harmonic and back with a speed that starts slowly (at zero), increases up to a moderate rate of speed, and then decreases to the original point. This creates a "shimmering" effect that suggested the title to me.
Each tone in the composition undergoes similar spectral changes, produced by moving a filter from one point in the spectrum to another, and back, in a varying periodic motion. The resulting effects sound different depending on where in the overtone spectrum the change occurs. There is always an interaction between individual overtones, which can be heard as separate tones, and both other overtones and the fundamental frequencies of other pitches in the same musical context. The rates at which the spectral changes occur are always in the subsonic range, and they are several octaves (nine or ten) below the fundamental frequencies of the notes on which they occur. The frequency continuum is thus structured on multiple levels: subsonically, in the rates of change of filters; in the mid-range, in the fundamental frequencies that constitute the composition's pitch structure; and in the high range, in the individual overtones that are emphasized in a periodic manner.
The piece is in six sections, each one evolving into the next. The opening states four all-interval tetrachords in four octaves that give the basic chords of the piece, and these are embedded in much of the remaining sections. The chords in this passage are a transposition of the voices.
The second section begins with a series of rapidly-modulated pentachords that are each stated in five descending octaves. These are followed by a passage that states the same chords in a contrapuntal manner, and it leads to a climax in section 3, a fast passage full of repeated notes, downward glissandos and crescendo-diminuendos.
The following section, 4, is a slow, contemplative passage that returns to the filtering, and the chords, of section 1. All the notes are stated in a single octave, but the filtering occupies four overlapping areas in the three octaves above the fundamentals. There are slow changes, but the speeds of the changes are different within each octave. In the second half of section 4, the chords and voices are interchanged. Section 5 is a slow climax of this material. Frequencies that had delimited the boundaries of the filters in the previous section are now introduced as pitches, and the notes are spread over a five-octave range. The filtering continues as before, only this time extending above the range of the highest notes on the piano. This leads to section 6, which continues the same filtering patterns but returns to the original tetrachords that have been used throughout. The notes in the lowest octave, brought into focus by the last note, is filtered from the fundamental up to five octaves above and back.
Timbre Study No. 5 was composed and synthesized entirely during my stay at the Gubbio 1991 festival in Gubbio, Italy. In the piece, the overtones of each tone are generated and controlled individually in order to create complex timbre changes. For each tone, a series of overtones is stated at the beginning of the sound that reflects the harmony of the surrounding passage. Tones are stated in basically three ways: (1) a cyclic pattern that states the overtone series at least twice over the course of the duration, (2) a complex envelope that states the series once with a changing timbre, and (3) a pattern that states each overtone individually, as a separate tone. Most sections use the first 16 or 24 partials to create the complete sound. Near the end, there is a passage that uses only high overtones, with no energy at the fundamental frequency and all the overtones concentrated in the same frequency area, as the fundamental frequencies reach into lower and lower octaves, until the entire series is introduced once again. Throughout the piece, there is a fascinating interplay and tension between the overtones and the fundamental frequencies or pitches produced by the series.
Meditation is my first completely microtonal composition, based entirely on 19-tone equal temperament. 19-tone temperament has been praised by theorists for many years because of its extremely well-tuned intervals, particularly major thirds and perfect fourths and fifths; these intervals are much more “in tune” with pure intonation than 12-tone equal temperament. On the other hand, major and minor seconds are more “out of tune” than in 12-tone temperament, and the basic 19-tone “minor second” is 63 cents, or about two-thirds of a semitone, producing some strange sounds. One thing that particularly fascinates me with 19-tone tuning is the fact that, for most pitch sets, new structures are produced by eighteen different multiplicative operations. These operations consist of expanding the intervals in a set by a given number of semitones, and they provide the most comprehensive method for incorporating pitch relationships in the system.
With the aid of a computer, I investigated the complete vocabulary of 19-tone pitch sets and developed a method of constructing arrays based on these relationships. These were the preliminaries that were completed before composed the piece itself.
Meditation, as the title implies, is a slow, contemplative work that begins from a single tone, combines it with other tones, builds to larger and faster materials, and ultimately returns to a single tone as in the beginning. The basic sound is a vocal-like tone produced by three-carrier FM synthesis so that two formants are emphasized. Throughout much of the piece, the sound undergoes a crescendo and diminuendo with a corresponding timbre change that parallels the basic structure of the piece. There is no amplitude or frequency modulation in the synthesis of the sounds; all the beating that is present is a natural result of the intonation of the tones.
There are five sections in the piece in a palindromic relationship and a 2:1 tempo change between each, increasing at first and then decreasing. In the beginning, tones start from the middle octave (the first note is middle C) and expand outward into other octaves. In the second section, where certain highlighted tones travel between the loudspeakers, the basic “theme” of the piece is stated. In the middle section, tones are attacked with a more "bell-like" envelope, and the exact midpoint is a climax. After that point, material returns in a compressed form, and the piece parallels the opening sections, returning to a single octave and single tone as in the beginning.
The piece was composed in 1993 and synthesized completely with the csound program.
Since no sound can be produced "spontaneously" by a computer except after considerable thought and programming, I must begin by explaining that I use the term "improvisation" to describe a piece that is spontaneously conceived, even though its execution takes a much longer time. These works may veer off in unexpected directions without necessarily returning to earlier themes. In Improvisation No. 2 the same two musical passages presented twice comprise the entire work: there are four sections, and the outer two and middle two present the identical music in different ways.
The unifying idea behind the piece is the overtone structure of chords -- the overtones of all tones in the chord considered together rather than as the sum of the four or five separate tones. This idea is also combined with glissando to create the impression of a constantly shifting sense of pitch. The piece begins with overtones unfolding so that only after several seconds can the listener actually perceive the chord, and as soon as this happens, the pitch begins to change and fade out. The next section introduces the overtones attacked suddenly in a bell-like fashion, with the pitch again moving as soon as it is established. The third section presents all the overtones together in a shifting pattern, but with each tone making a slow glissando just after the pitch is established. A climax occurs as all tones coalesce to a single pitch, which then dissipates. The final passage presents a single-tone "drone" against the high overtones of the other notes of the chord moving in a separate rhythm. At the end, only the "residue" of the chord remains as it dies away.
Improvisation No. 3 is another of my pieces based on the sounds of natural overtones in changing patterns and unusual ways. In addition to using the overtones of chords as a single unit and cyclic variations of the overtones of a single tone, in this work I have created a type of "bell" sound by "squashing" the first sixteen overtones into the space occupied by the first fifteen, producing inharmonic relationships among the components. Only the first and fifteenth overtones are "in tune".
The piece begins with a big crashing bell sound, which presents all the harmonic content of the piece simultaneously, with both components and chords decaying at different rates. Then a slow, low passage emerges which gradually builds as it moves into higher octaves. This is followed by a passage combining middle- and low-octave glissandos with much higher overtones of the same chords descending downward over a six-octave span. The climax of the piece presents the bell sounds against constantly-shifting chords. The conclusion combines all of these materials into one mosaic spanning eight octaves: high chordal overtones drifting downward in patterns, middle-register bells, and a low constantly-moving "drone" with overtones extending up five octaves into the bell sounds.
I have always been fascinated with the process whereby the overtones of a sound, which are in fact separate tones, fuse into a single pitch, with the fundamental frequency being the pitch perceived and the overtones being perceived as the timbre or tone color of the sound. All of my Timbre Studies deal with this idea in one way or another.
This composition is basically a study in filtering and glissando. It originated from the fantasy of listening to the upper overtones of a sound and filtering them sharply to focus on a specific pitch area. These areas are then structured as pitches themselves, with a similar type of content to the lower notes that provide the overtone material. This fantasy is realized explicitly in the middle section, where one hears a low passage that proceeds on its course and a higher, faster passage that creates a counterpoint to the slower passage in the filters. As for the outer sections of the piece, an analogy is created between fixed pitches as relating to fixed filters and variable filters to glissandos. Contrast the variable filter (which creates a sort of "wa-wa" or "yeow" effect) against the fixed pitches at the end of the first section with the glissandos against the fixed filters in the beginning of the last section. Overall, the piece is a large crescendo-diminuendo, beginning and ending on a single note, growing and shrinking along the same lines. The frequency content that is structured in the piece exceeds that of the piano keyboard, from the lowest note to higher than the highest note.
Improvisation No. 4 is the fourth of a series of pieces that deal with the overtone structures of sounds, as well as vibrato and glissando, in unique ways. In this work, each tone in the upper octaves consist of eight partials, in the middle 16 partials, and the lowest 24 partials. The overtones are always generated individually and in ascending or descending order. The overall shape of the piece is a palindrome, starting at the very slow tempo of 10 beats per minute and increasing (usually doubling) the tempo of each successive section until the middle, which reaches the tempo of 120 beats per minute, and then reversing the process.
In the opening section, the overtones are stated in a manner that "unfolds" the tones: the highest overtones enter individually one at a time, and the sense of the pitch of the tone does not enter until several of them are sounding together. They decay in a similar way. After an initial delay, the pitch of the tone begins a very small vibrato, the speed of which is harmonically related to the fundamental: it is eight octaves below (in a subsonic range).
The second (and penultimate) section develop what I describe as a "dissolution" of the tone. The overtones enter as a group, and then individually begin a glissando to the corresponding overtone of the next note. There are four lines, each of which trace a separate path, and some of the glissandos are very long and cover a small interval while others are short and cover much wider intervals, although each line exists within a separate octave.
The third and fourth sections use an instrument that produces all overtones together but with increasingly longer attack and decay times, producing a "swelling" effect that simultaneously makes a crescendo and diminuendo. Once the pitch is established, a small vibrato like that of the unfolding sound enters. In the fourth section, the overtones of the sound make a slight glissando up and down rather than the vibrato. As the section proceeds, the amount of glissando increases.
The middle of the piece (actually comprising three sections in length and taking two minutes) introduce a very low note, G in the lowest octave of the piano, that, over the course of the complete passage, makes a downward glissando of an octave, to G off the end of the piano. The entire music played here makes a similar downward glissando of the same interval, which counteracts the motion of the three sections themselves, which move up an octave over the passage.
After the mid-point of the piece, the instruments and tempos of the first sections return in reverse order, except that the material is compressed and occupies fewer measures. The climax of the piece occurs in the next to last section, after which the final section unfolds its tones as a sort of coda.
The piece was composed in 1998 and synthesized entirely by the csound music synthesis program.
Cacophony is a work based on instruments that generate overtones individually and in different ways. Tones in the octave of middle C and below use 32 partials, spanning a space of six octaves above the fundamental. Tones in the octave above middle C use 16 partials, and those two octaves above use 8 partials. The music is constructed so that sound usually fills the entire spectrum from the lowest note on the piano (some notes go below that) to above the highest note. The title appropriately reflects the depth and breath of the sounds in the piece.
One group of instruments unfolds the harmonic partials of the tone in an ascending order, another group does so in a descending order, and a third group does so in an uneven manner, so that the opening of the sound states the "harmony" of the sound of the context in which it appears. Partials 1, 2, 4, 5, 6, 8 and their higher octaves state a major triad, for example; but for more complicated chords, the partials above 16 can state any harmony, no matter how complex. Since these are harmonic partials, their statement is in pure intonation, even though the music is in equal temperament. (Several partials are unusable in this way because they are too much out of tune with notes of the tempered scale.) Beyond this idea, there are different types of dynamics, processes, and envelopes applied to the sounds. One instrument repeats the tones where each pitch has a unique speed. Another instrument plays shifting cycles of the overtone pattern, and a different one states the overtones in a complex envelope. The "shimmering" sounds heard in the beginning and ending sections unfold the overtones of a chord in the low registers using overtones several octaves above the fundamentals.
The music is based on a series of interlocking array structures including a controlling array and two others, so that the entire series states the total chromatic and each has certain unique properties. This entire structure is only presented in the last section; in the earlier passages, each successive section includes a restatement of one of the elements in the preceding section, but played by a different instrument, so that its identity is not always immediately apparent. The restating idea gives a sense of continuity and progression. There are seven total sections. The tempo begins at 15 beats per minute doubles in each section up to the mid-point; afterwards, it is halved to return to the opening tempo at the end. Many of the passages include two statements in different time signatures.
One of my concerns in computer music is the development of interesting, original sounds. The sounds in this piece were conceived by imagining what would happen if the octave were replaced by something different. The ratio between the frequencies that are an octave apart is 1:2. In this work, the 1:2 octave ratio is replaced by the value of the square root of 2 and the square root of 3; the next number in this series would be the square root of 4, or 2, which is the usual value. (This idea was suggested to me by John Chowning’s piece Stria, in which he employs an analogous sound derived from the golden mean of 1:1.618.)
Instead of the usual harmonic series, the overtones in this work exist in the ratios of the square roots of 2 and 3. For the square root of 2, the series consists of the octaves and the square root of 2 values in between: 1, 1.414, 2, 2.828, 4, 5.657, 8, 11.314, 16, 22.627, 32, 45.255 and 64. For the square root of 3, the partials are in the ratio 1, 1.732, 3, 5.196, 9, 15.588, 27, 46.765, and 81. (Not all sounds use the entire series; when the upper partials are above the range of human hearing, they are not generated.) In addition to using these harmonic series, the work also derives 12-tone equal-tempered scales from these values. For the square root of 2, the scale is quarter tones, but the square root of 3 produces a more unusual series where 12 steps fit into about a major tenth and the step is about 5/6th of a half step. Within a usual range of frequencies less than the span of keys on the piano, there are 12 octaves in the quarter-tone sections and 8 in the square root of 3 sections.
In addition to these sounds, the piece also employs frequency-modulated sounds using the ratios of 1:1.414 and 1:1.732. These sounds can be described as “bell-like” and are employed with a “crescendo-diminuendo” spectral envelope that changes the timbre, whereas the square root tones unfold the partials in an upward direction.
The piece is in four continuous sections with a palindromic structure. The first half uses the square root of 2 and the second the square root of 3. It begins slowly with the tempo accelerating to 10 times the original tempo, then holding that value through the middle and decelerating after the middle. In the mid-point of the piece, where the transition between the square root of 2 and 3 occurs, there are two passages that are identical except one is based on the square root of 2 and the other on the square root of 3.
Mosaic employs filtering of the overtones of the sounds that occur. There are several different ways in which the computer "instruments" that perform the music do this, but all use similar principles and methods. The piece begins with tones that "unfold" the overtones in an upwardly ascending manner. Tones in different octaves have more area in which to operate, so they naturally ascend to higher overtones. The piece begins (and ends) with a single tone, after which more complex sonorities unfold. We hear each of the overtones in the series enter as the emphasis rises, and they interact with the other tones in the context. The second and, in some ways, most important instrument employs three resonances that oscillate above and below their mid-points, thus producing a kind of diphthong effect. The speed of these oscillations increases and decreases over the tone's duration. This instrument is used throughout the middle section, although it is combined with others. The third instrument employs the same three resonances as the second, but instead of oscillating they descend at different speeds to the fundamental, thus "dissolving" the sound into the fundamental. The fourth instrument, used only in parts of the middle sections which play "chorale" passages, simply unfold the overtones in a manner similar to a brass instrument, which introduces gradually higher overtones as the tone increases in amplitude.
The piece is based on a series of interlocking arrays that produce a cycle that exhausts all of the possible combinations in which a particular collection of tones (the octachord that excludes 02350) can be generated. The arrays are all based on either trichords and pentachords, and the interlocking manner in which the trichords are imbedded in the pentachords is what suggested the title to me. In each passage, vestiges of the previous passage appear, sometimes in the same and sometimes in different rhythms.
The piece was composed in the Fall of 2000 and synthesized by the csound program.
LongGong is based on the sound of the same name that is one of the presets on the Yamaha DX7-II synthesizer, which I have enjoyed since I first heard it. It consists of three sine-wave carriers and one modulator that controls two carriers. The frequencies (ratios) of the three sine waves are 1.0, 1.42 and 2.14, and the two modulators create c:m ratios of 1.21:1.75 and 0.76:1.75. These processes create a series of which the first 22 elements are the partials .54, .76, .99, 1.21, 2.29, 2.51, 2.74, 2.96, 4.04, ..., 9.96.
All qualities of the composition, including both pitch and time elements, are created from the elements of this sound. The piece begins with a single tone that states the basic LongGong sound, followed by an echo that transposes the frequencies to each of the elements to the series. The time values are the values of the series transposed to seconds. Following this, the entire remainder of the composition is a fractal based on this sound. The elements of the series are transposed to minutes, and a complete series of tones that plays the entire series transposed to each of the elements of the series, in a slowly increasing series of time values, changing from 9.96 minutes to 9.96 seconds. Thus, the piece grows in intensity and frequency towards the ending. There is no chromatic scale, nor are there any traditional rhythmic values. It is my personal homage to this sound.
Both of my compositions entitled "Cacophony" deal with sounds that cover a vast stretch of frequencies, from below the lowest to above the highest notes on the piano. In this work, based on filtering, there are two different structures of pitches and rhythms. At the low end are the source tones, which provide a continuously evolving musical passage of between three and five octaves, and at the high end are the filters, which operate on the overtones of the lower sounds and provide their own evolving musical passage, often with different rhythms and pitches from the source tones. For much of the composition, there are independent pitches and rhythms structured over an eight-octave range.
There are several kinds of filters that are used in the piece, all of which fall into two basic categories: fixed and variable. The fixed filters resonate specific areas of the frequency continuum (specified and distinguished as pitches), but the only variable is the amplitude of the filter. Many passages alternate between four different fixed filters, fading in and out between them. Another type of fixed filter is the complex envelope, in which a number (four in this piece) of resonances is attacked within each note each with a different envelope. The variable filters change in frequency over time, producing effects sounding like the diphthongs "wa-wa" and "yeow". In this case the important variables are the time span of the change and the amount of frequency covered, which ranges from less than an octave to three octaves. Sometimes the variable filters are keyed to move through specific overtones of the fundamental frequency, and at other times they articulate a separate structure from the source tones in their own rhythms.
The piece is in twelve sections with the overall shape of a palindrome. The outer sections are the slowest, each lasting over three minutes, and the tempo doubles five times as the piece progresses from the beginning to the middle and is cut in half five times from the middle to the end (the fourth and fifth, and correspondingly the eighth and ninth sections are in the same tempo). The music proceeds by inclusion, whereby parts of the preceding passage are incorporated into the next one, and so on. The music in the two halves of the piece are related by cycle-of-fifths equivalence.
The piece was synthesized in January and February of 2002 using the csound program.
Iridescence was initially composed in November 2002 in response to my friend Dinu Ghezzo's request for a piece to be included on a concert he was planning, and revised in April 2003 . It is based on sounds that employ both fixed and variable filters that produce many different kinds of colorful "shimmering" effects, which suggested the title to me.
The piece is based on a succession of two sets of arrays (excluding the sets 0135(5) and 0356(4)) that include common elements and complement each other, so that successions and combinations of them produce unique properties. It begins slowly in a single octave and expands outward in stages, ultimately building to a climax that spans a 7-octave range (the entire range of the piano keyboard). After this, a series of shorter passages emerge that make the transition from the first to the second group of pitch materials, and the piece progresses in a quasi-palindromic but compressed fashion to the ending. Many sections of the piece repeat literally some of the music from the preceding section, integrating that material into a new context. Whenever this occurs, the repeated material is reverberated.
The tempo of the piece, while starting slowly, accelerates to four times the original tempo in the middle and ends in a faster tempo that it began.
All tones in the piece employ filters that are in the range of three octaves above the fundamental (except for tones in the highest octave of the piano, where they are up to two octaves above). These resonances articulate a different set of pitches from the fundamental, but a series that is related to the overall passage. One group of instruments employs variable filtering (this is part of the "iridescent" quality of the tones), while another uses fixed filters and variable amplitude modulation. Detuning among components produces a kind of chorus effect, and there is also a very slight variable vibrato. All speeds of these variable qualities occur at subsonic frequencies in ranges of eight to ten octaves below the fundamental.
Harmonic Fantasy is based upon very rich sounds, all consisting of 32 harmonic partials, which extend five octaves above the fundamental (except on high tones, where the partials exceed the limits of human hearing). The harmonics are introduced one at a time in an irregular series that emphasizes the harmony of the context in which the tone appears at the beginning of the series, followed by a transposition of the series, and finally by the remaining partials. Following the introduction of the individual partials, the tones undergo either vibrato or glissando in precisely controlled ways. Vibrato is applied to the partials in an individual, out-of-sync fashion at a subsonic speed that is seven octaves below the fundamental (thus, middle C would be about 2 Hz). The partials of glissandos are also delayed by a distinct amount and move individually to the corresponding partial in a new tone. This creates the effect of the sound dissembling before your ears, only to re-coalesce into a new tone. In the second section of the piece, these tones create a three-part melodic context, but in the later sections where these are used, the tones move up a minor third and back to the original tone over the context of the tone's duration.
The piece is in six sections, beginning with a thin texture of trichords and building by accretion to more complicated harmonies and textures. Each new harmony is formed by adding one tone to the chord from the previous section, until a hexachordal texture is reached. The piece grows dynamically in a manner similar to Ravel's Bolero, reaching a huge climax in the fifth section. The concluding sixth section extrapolates three-note chords from this passage into a new structure and concludes softly.
Harmonic Fantasy was commissioned by Winthrop University in Rock Hill, South Carolina. It was sketched while I visited Singapore in October, 2003 but could not be produced until I returned home.
In composing this work, I sought to create a new kind of sound containing clusters of non-harmonic partials, all compressed into a small span, and to combine that with the idea of a musical fractal. I composed a musical passage which develops according to its own logic, divided into twelve sections. Different sections span three, four and five octaves. I then created a series of computer instruments in which each note plays a complete passage consisting of all the notes in its own section, as sine tones (inharmonic partials), squeezed into the interval of a perfect fifth. Each of the tones enters in the same rhythm as the notes in the complete passage, the only differences being that the sequence is transposed to the level of the pitch in the score, and all of the partials take place within the time-span of the duration of the note. Different sections consist of 12, 28, 30, 48, 56, 64 and 68 components.
The work was composed in 2005 and synthesized with the csound program.
Inharmonic Fantasy is based on the idea of squeezing the harmonic partials of a tone into a smaller interval, where the tone loses its sensation of pitch but still retains a distinct identity. The piece begins with a single long tone, with harmonic partials fading in and out in a pattern; but before long, they “dissolve” through glissandos into a sound that has no real pitch. The partials are always equally spaced, and the fading process always keeps going. As the piece continues, partials are compressed into smaller and smaller intervals, ultimately squeezing the five octave range of the first 32 harmonic partials into the space of an octave and a fifth. In the middle section of the piece, tones are attacked like gongs, and while they do not make glissandos, successive tones are further compressed. Glissandos return at the end, where the piece closes out with harmonic partials, as it began. The piece was synthesized with the csound program.
In computer programming, a macro is a series of codes that can be defined in a prototype and then called in just one line. In Macro Structure 2, each note brings forth a series of notes that duplicate the harmony of its context. The piece progresses from sections based on trichords through tetrachords, pentachords and hexachords by adding one or more notes to the preceding chords, and then reverses the process. Each measure occupies the same duration of ten seconds, but successive sections subdivide the basic duration into a greater number of beats, so the tempo appears to increase and then decrease. Each tone consists of harmonic partials that enter at progressive delays and are sustained for half the duration, then drop out in the same order. In the stereo version, each tone begins in a specific location and then moves to one loudspeaker and then to the other, finally returning to the same location where it began. Notes at the edge (located entirely in one speaker) are completely dry (unreverberated), but as they move to the center, the reverberation increases, partly creating the impression of the note receding into the distance (only partly because, if the sound did move in physical space, it would get softer). In the quadraphonic and octaphonic versions of the piece, the center front location is the only dry spot, and reverberation increases as the sound moves to the back.
Inharmonic Fantasy No. 2 is based entirely on sounds containing inharmonic partials, that is, overtones that do not so much create a timbre for the sound as they create a kind of cluster above the fundamental. The overtones have the same relationship, entering in the same rhythm and in the same pitch relationship, as all of the notes in the passage in which they occur. Thus, the sounds themselves are a kind of fractal. There are two ways in which they are presented: continuously, as a kind of complex envelope, and attacked as separate tones. Underlying everything is a very slow vibrato that expands from zero to a perfect fourth in the middle of the piece, making just one cycle over each entire section.
The piece opens with a soft and slow passage that unfolds the basic idea, and which underlies most of the piece, except for the middle section. This is followed by a faster and denser passage, and then a less dense but even faster passage that introduces the attacking tones. There is a short pause in the middle of this section, after which a climax occurs that uses both instruments together. After this reaches its apex, the underlying tones similar to the beginning are left, and the piece ends quietly.
The work was composed in 2007 and synthesized with the csound program.
Groans is a slow work based on a single sound, its spectral inversion, and a third timbre that is a combination of the two. The basic sound is a tone in which 32 harmonic partials, spread over six octaves, emerge one at a time in an ascending series and reach asynchronous peaks at the same time as they increase greatly in amplitude, and then fall back in the same way. The title is a description of my first impression of this sound, as well as the audience's expected reaction to it. The spectral inversion is a sound in which the same series of partials descends from the top of the spectrum to the bottom at the same time as they decay, making a crescendo at the end of the tone. Some tones also undergo variable amplitude modulation. The piece begins very softly in a single octave, then expands to two, three, and more. After a short time, a second strand of music begins and develops the same way, the texture increasing in each successive section. At the climax of the piece, the inversional instrument enters and dominates the music for two of its nine sections, during which time there is an acceleration and deceleration, after which is subsides, and the piece ends in the manner in which it began. It was synthesized on my home computer using Csound.
Timbre Study No. 7 is based on overtone patterns, clusters, and the squeezing of the harmonic spectrum into smaller intervals. Each tone consists of a series of separate pitches that expand the harmony of the surrounding context. Separate passages are based on trichords, tetrachords and pentachords. There are, appropriately, seven sections, which are all (except the sixth) the same number of beats, but the tempo accelerates in the middle and slows considerably in the end. The first unfolds 32 harmonic partials in an ascending manner, but in an overall downward pattern. In all the remaining sections, these overtones are unfolded in a pattern that first plays the harmony of the passage, beginning from the sixteenth partial, then a transposition of the harmony, and then all the remaining partials. For example, the major triad (047) would begin with partials 16, 20 and 24, then 32, 8, 10, 12, and so forth. The advantage of starting from the sixteenth partial is that all intervals of the chromatic scale are available beginning from that tone, in just intonation. In the third section, overtones are progressively squeezed down to the seventh partial, and the fourth and fifth sections consist of these “scrunched” tones, and in the sixth they expand progressively outward to the harmonic series again. The final, longest, section is based entirely on trichords, with tones fading in and out, dwelling on the rich sounds that are at the heart of this work.
Accretions is my second piece in 19-tone equal temperament, and it shares many qualities with my first piece, Meditation, which embodies my approach to writing in this temperament in general. Basically, I strive to express a sense of harmony in which each melody and chord that sounds is related to all the others in its context through multiplicative operations. Each passage is built from a sequence of chords that states all 19 notes and also contains all the 19-tone intervals, except in passages that are built from fewer notes than would make this possible. Multiplicative operations expand the intervals of a chord into greater degrees of separation, and since 19 is a prime number, all these operations are unique, related otherwise only to their complementary operations (inversion). Successive passages are built by adding a note to each of the previous notes in the chord until it reach seven, at the climax of the piece, which also covers seven octaves, nearly the full span of the piano keyboard. After reaching this point, the piece contracts to its original configuration, except that passages are compressed, alternating thin and dense measures, where the notes are more concentrated in a single octave. To allow the listener to focus on the intervals, all notes are stated plainly, without vibrato or tremolo; all beating is a result of the tuning system. The piece was generated by the csound program.
Most sounds that we hear in music consist of a spectrum of harmonic partials or overtones, and sometimes these also include some inharmonic components. In Clusters,, the overtones are all clusters of 5-note chords duplicated through three to four octaves above the note. In other words, harmony becomes spectrum. The amplitudes of these components are varied so that they have a kind of “shimmer” moving up and down the spectrum. There are five different kinds of “instruments” used in the piece: the basic cluster, a “variegated” cluster, a “whoosh” sound that attacks each of the components separately, a “gong” sound, and a cluster glissando. The piece begins in the middle range and proceeds through several short passages, each emphasizing a different aspect of the sounds, until in reaches a big climax with all instruments being used, and finally concludes quietly, much as it began. The piece was synthesized using csound.
19-tone Clusters takes the same assumptions that went into my composition Clusters and applies it to the domain of 19-tone equal temperament. All the overtones are all clusters of 5-note chords duplicated through three to four octaves above the note, but they are all in 19-tone equal temperament. Again, harmony becomes spectrum. The amplitudes of these components are varied so that they have a kind of “shimmer” moving up and down the spectrum. There are five different kinds of “instruments” used in the piece: the basic cluster, a “sparkling” cluster, a “whoosh” sound that attacks each of the components separately, a “gong” sound, and a cluster glissando. Consistent with my theories of 19-tone music, each short passage is based on different but related chords, and passages state both the entire 19-tone pitch classes and all nine interval classes. The piece has the same overall structure as Clusters, beginning in the middle range and proceeding through several short passages to a big climax with all instruments playing, and finally concluding quietly, much as it began. The piece was synthesized using csound.
When the microsound mailing list announced their annual Pi day celebration, I was intrigued. I thought about the number of ways in which an abstract number like Pi could be incorporated into music, and I came up with a number of interesting ideas.
The first was to create a scale based on the number Pi. As you know, equal-tempered tunings are based on the number 2, since 2 is the ratio between octaves; the formula for a half step is to take the twelfth root of 2 and raise it to the power of the octave (e.g., 8.25 is the exponent for 2 for the note E-flat or D-sharp in the 12-tone scale). So, instead of using 2, I used Pi as the base of the scale, and divided it into 12 steps. There aren’t any of the usual octaves in this scale, since the ratio of Pi becomes the “pseudo” octave. Within the range of audible frequencies, there are about four usable octaves of this scale, stretching from 31.01 to 2745.51 Hz. The next step was to use Pi as the basis for determining overtones of the sounds. Harmonic partials are integral multiples of the fundamental frequency. For these sounds, there are nine partials at integral multiples of either Pi itself or 1/Pi, so they go from 3.1416 to 31.01, or from .318 to 2.86 for the inverse. Having finished these steps, I then tested several notes based on these assumptions.
The composition Pi itself is exactly 3 minutes and 14.16 seconds long. The work is in four sections. The first introduces the sounds, the second expands on them, the third uses the inverse sound, and the fourth returns to the texture of the second. Like all these compositions, the piece was generated by csound.
Emergence is based upon the fascinating thing that happens when a group of independent tones are played together, and when they are tuned in a precise manner, so that they are in a harmonic relationship, another note jumps out – the fundamental – and now we hear only that second note, and all the others are taken in as the timbre of the sound. All notes in the piece are created with up to 32 harmonic partials, and they are presented in three ways: as independently attacking tones, as continuously fading tones, and as a complex envelope. All notes are played with a pattern of overtones that begins from the sixteenth partial and state the “harmony” of the context in which the note occurs. The harmony is clearly discernible at the beginning of the sound, but it later merges into the timbre. The work concentrates on the interplay between the overtones and the fundamentals they are a part of.
Inharmonic partials are sounds that are not the overtones that we hear with most instrumental or vocal sounds because they do not combine to create a sense of pitch. Another way of describing them is that they are sounds that have a spectrum but not a "timbre" in the way that we usually think. This work was conceived from a desire to create complex, evolving inharmonic sounds that include many different components that fade in and out over the course of a tone. The sounds were created by combining the pitches that occur in many different octaves and compressing them into the interval of an octave and a fifth, or a twelfth. The work consists of numerous short passages that include different numbers of notes, densities, and rhythmic distributions. The inharmonic components are presented in ways that both fade in and out over the course of the tone or are attacked and decay separately. At the climax of the work, these two processes are combined. The piece was a commission from Nancy Bogen, written in 2014, and synthesized using csound.
Inharmonic partials are sounds that are not harmonically related to each other, as they are in most instrumental or vocal sounds, because they do not combine to create a sense of pitch. This work is another in the series of pieces I have written in order to create complex, evolving inharmonic sounds that include many different components that fade in and out over the course of a tone. In this work, the sounds are all compressed into the very small acoustic space of less than a perfect fifth. While each sound occupies only that small area, the tones within each passage sometimes are also compressed within a small space, or are spaced widely over the acoustic spectrum. The work consists of numerous short passages that include different numbers of notes, densities, and rhythmic distributions. The inharmonic components are presented in ways that both fade in and out over the course of the tone or are attacked and decay separately. The piece was written in 2015 and synthesized using csound.
Inharmonic partials are sounds that are not harmonically related to each other, as they are in most instrumental or vocal sounds, because they do not combine to create a sense of pitch. This work is the fifth in a series of pieces I have written in order to create complex, evolving inharmonic sounds that include many different components that fade in and out over the course of a tone. In this work, the sounds are all compressed into the acoustic space of two octaves and a perfect fifth. The tones within each passage are spaced widely over the acoustic spectrum, usually three or five octaves. The work consists of numerous short passages that include different numbers of notes, densities, and rhythmic distributions. The inharmonic components are presented in ways that both fade in and out over the course of the tone or are attacked and decay separately. The piece was written in 2015 and synthesized using csound.
Expansions is so named because each note in the basic framework is “expanded” by another group of notes. Textures are presented so that the expanded notes both fade into a complex envelope or are attacked individually. All notes are tempered pitches. The process gives rise to complex harmonies, but no inharmonic components. The piece was composed in the summer of 2016 and synthesized by csound.
Like my other inharmonic fantasies, this work consists of tones with different components that fade in and out over the course of the duration. In this work, however, all the components are quarter tones, or notes that are squeezed between the half steps of the 12-tone tempered scale. The background structure of the piece is based on tempered pitches and can be perceived on the entrances of the notes. The components both fade in and out or are attacked individually. The piece was synthesized by the csound program in 2018.
Unbalanced is my first work for a solo instrument with electronic accompaniment. The piece is in four large sections, with a fast introduction, a slow middle section, a fast quasi-reprise of the beginning, and a moderately paced ending. In it, I have tried to take advantage of the expressive qualities of the saxophone against the fixed aspects of the electronic part, which are also expressive in their own way. The piece consists of numerous short sections that make up each larger section, and some of the short passages feature many notes in one octave against a few in others. The saxophone is almost always playing alone in the register where it occurs. The “unbalanced” nature of this writing is what suggested the title to me. The piece was written in 2013-14, and the electronic part is generated by the csound program.
The basic idea behind my Harmonic Fantasy No. 2 for Piano and Electronic Sounds is that the fixed media part, generated by computer, resonates and complements the sounds of the piano. There is never any opposition, or even any counterpoint, between the two forces in the work. There are four sections to the piece, which are like four separate movements. In the beginning, the piano plays a few notes, and the computer sustains them, while emphasizing certain upper partials in a repeated pattern. This idea continues in the second section, which is a slow passage that builds increased complexity through accretion, by adding more notes to the passage that it began with. The third section begins when the piano starts playing fast scale-like passages in the higher register. These notes are actually the same resonances that have been used in the opening, only now the computer plays the lower fundamentals to which these are the resonances. These roles exchange in the last section, where the piano returns to playing low, sustained tones while the computer resonates the upper harmonics. The term “harmonic” does not refer to harmony in the traditional sense, but to overtones, which are the focus of the sounds. The score shows only the fundamental frequencies, not the resonances of the upper partials which are such an important part of the piece’s surface texture. The position of notes on the staves in the fixed media part is not significant; they are only a convenience that enables the notes to be written in the most economical manner. The piece was commissioned by Nancy Bogen and is dedicated to her and to her husband, Arnold Greissle-Schoenberg.
My Inharmonic Fantasy No. 6 develops sounds similar to what I have employed in my other inharmonic fantasies. In all these works, the sounds are undergirded by simple melodies which are harmonized with inharmonic elements. In this work, the flute plays the underlining melody explicitly, while the computer plays the inharmonic sounds. There are two basic kinds of sounds that are used in the piece: those in which the components fade in and out over the course of the tone and those in which the components are attacked individually. These are represented by the instruments that are designated “1" and “2" respectively in the score. The piece begins with the first instrument and then introduces the second, and at the climax both are playing together. The score does not show the durations of the notes in the fixed media part, except at the beginning and ending; it shows only the attack times and pitches. The score also shows only the underlying notes, not the inharmonic components, of which there are several for each note shown. The flute should be miked and mixed with the fixed media part so that both always sound at a balanced level. The dynamic levels range from piano to fortissimo, so this may require some adjustment during the performance.
Piece for DX-7 II Synthesizer Ensemble was written in 1988, after I had just acquired a Yamaha DX-7 synthesizer. I became intrigued by the sounds it could produce, not just the imitations of musical instruments, but particularly the interesting and unusual "bell-like" sounds it could create by using simple fractional ratios. This work is a reflection of my obsession with one of those sounds, called "BellWahh A" by the manufacturer. The attack of the sound is like a sharp bell, but as that component decays, another vocal-like component enters and builds slowly. I found that this sound worked particularly well with the type of harmony that this piece uses, in which complex chords are built up slowly. Each tone has components in at least two octaves, and one of the sounds near the end, a low D at 18.32 Hz, is the lowest sound I have ever seen used in a piece of music.
The work has now been synthesized entirely by the computer.
Procession is essentially a slow march. I envisioned describing a slow-moving procession moving past a fixed location with an unrelenting motion toward a foreboding destination. The work is in three parts, with the faster midle section at twice the tempo of the slower outer sections.
The piece is scored for three synthesizers in live performance. Two of them play a sustaining timbre, with a rich sound containing many harmonics. The original sound I used was a mix of stringlike and brasslike timbres, but any rich sustaining sound will do. The other synthesizer plays a sustaining tubular bell sound, rather like a tuned chime, but extending over a much greater range than conventional chimes.
The piece was written in Tuscaloosa, Alabama in 1989, when I was a visiting professor at the University of Alabama.
My Piece for Five-Octave Keyboard was written in 1985 and 1986, at a time when I was beginning to be enthralled by the possibilities of live performance on the new MIDI keyboards, particularly the Yamaha DX-7. These instruments were the first that offered the opportunity to play music with the same range of expression as acoustic instruments. One of their main limitations was that the keyboards were limited to five octaves or 60 keys, unlike the 88 keys on the normal grand piano. Therefore, this became the main limitation of the piece. The timbre to be used didn’t really matter that much to me, as long as the sounds had decaying characteristics similar to a piano. In fact, I rather liked the possibility that the tones could be sustained a little longer than on the piano, but only marginally. Now, however, I feel that the piece is best played either on a normal grand piano, or on an instrument with weighted keys and the same sort of feel to the pianist as a piano. The DX-7 is usually placed on a flimsy stand in performance, and it or the pedals can easily become knocked around in a performance as physical and virtuosic as this piece demands.
The above statement was what I wrote for the original version of this work, which was for a single performer. After hearing a couple of dedicated pianists attempt to perform the work, it became clear to me that it was impossible to do so in the way I had written it, because some of the stretches called for were simply too large. This problem is corrected with this new version for a single instrument, two players. While there are some passages that can easily be played by one player, the difficult passages at the beginning and end are clearly better with two, and I believe that this version is playable.
The work is a palindromic structure, beginning with a full maestoso-like statement which leads to a frantic passage that is to be played as fast as possible, the only time I have ever used such a directive (and it returns before the ending). There is then a long transition which slows and compresses to a single octave, followed by a quiet and introspective middle section. In the mid-point of the piece, there is a faster passage in duple meter that has a jazzy-like rhythm, followed by a return of the other passages in reverse order. These returning passages are actually strict transformations of the previous ones in the opening sections.
I have reconstructed this work rather than simply let it lapse because I think it is important in my overall development as a composer, when I first developed some of the ideas that I have employed subsequently in my other works.
The opening passage of my Quintet (8 measures) states a succession of chords (the four all-interval tetrachords) from which the entire composition is derived. Each subsequent passage in the work relates directly to these chords, in the same order in which they are stated here. While there are two movements, the work is a single unit with a palindromic structure. The beginning of the second movement is the exact midpoint, and the succeeding passages relate, in reverse order, to the first movement.
The basic tempo of the piece (first stated at ) is 60 beats per minute, four beats of which occupy four seconds. Divided into sixteenth notes, there are sixteen ticks in a measure. Other tempos relate to this by dividing the same four seconds into different subdivisions: 12 yields a tempo of 45 beats per minute (60 times 12 divided by 16), and 20 yields 75 (60 times 20 divided by 16). The piece begins at the slower tempo, and the main body of the first movement is at the faster tempo; the median tempo is used in transitions.
The opening passage ( and ) expands the original chords into several octaves with increasing complexity and rhythmic interaction between the instruments. At , a transition occurs at a faster tempo that states the basic materials for the main body of the first movement. At , the tempo increases again, and there is a lengthy passage expanding and developing this material. At , a softer and less intense interlude affords the opportunity for solos on each of the instruments. This leads to a climax at , which is the loudest and most complicated section of the piece. When this has finished, the tempo slows, and a soft conclusion () functions as a transition to the second movement.
The second movement begins with a slow, lyrical passage () based on trichords. This is the purest and simplest passage in the entire work. It is followed by another passage () only slightly more complex. There follows an extended development ( and ) that alternates loud and soft passages and which begin to retrace the steps that led from the beginning of the piece. At  and , the tempo changes to that of the end of the first movement, followed by an even faster passage ( and ) that corresponds to the middle of the first movement. At , the tempo slows to that of the first transition at , the material from there being stated in compressed form. The ending retraces the material from the opening passages all the way back to the beginning, gradually diminishing and contracting until just a single chord is left.
The basic material of the piece consists of the four chords stated at the beginning, which are related by certain basic structural operations, and these are the only chords that exist with various specific properties. These chords are fragmented (the opening passage states them as a single note followed by a trichord) and combined with other notes to form new chords that usually possess similar unique qualities. In some later passages, successions of four different chords related by the same operations which state or present each of the original chords in some unique way are used, thus producing lengthy passages of considerable complexity all derived from a single chord.
My Nonet is in two movements, approximately 15 and 12 minutes respectively. The opening of the first movement, three phrases based on groups of 3, 4 and 5 notes, states the materials on which the entire piece is based. The first movement is a kind of sonata form based on two contrasting ideas. The first, presented in 5/4, begins immediately after the slow introduction. Several contrasting passages build to a climax and lead to the second idea, presented more slowly and softly in 4/4. After a slow transition, a lengthy development ensues, in which these two ideas are combined in different ways, along with some digressions. The development continuously intensifies up to the recapitulation, which presents a cycle-of-fifths transformation of the original idea, while the developmental processes (loud interruptions occurring within softer passages) continues. These processes extend through the return of the second idea, leading to a slow and rather lengthy coda. The ending summarizes all the materials presented in the movement.<.p>
The second movement is a very slow series of excursions that are spun out of the opening duet between the oboe and violin. The opening intensifies very gradually as new elements are added in each successive phrase. Finally a faster, louder passage begins, interrupted by silent punctuation marks. This passage wanders through many different areas while leading to quieter, more intricate harmonies. These "wanderings" are nevertheless connected by a common thread. After a brief return to the opening texture, a melancholy recapitulation of the opening occurs. As in the first movement, this is a cycle-of-fifths transformation. This leads to a slower transformation of the second section, which moves through a similar series of excursions. The conclusion recalls the opening of the first movement, stated so that it recalls the entire piece.
The overall shape of my Chamber Concerto is a sonata form, with an introduction, two main thematic groups, one presented in 5 and the other in 4, a development, recapitulation, and coda. The introduction consists of two brief phrases, the first based on trichords and presented in the string quartet, and the second based on pentachords and presented in all instruments except the piano. This passage is restated several times in the piece at structurally important points. The opening four measures are completely integrated into the following four, establishing a connecting principle that is used throughout the piece.
The first theme begins at 2, stating the main idea, which is a series of pentachords connected by trichords. A brief transition connects to the second theme at 4. This leads to a brief digression on the winds, followed by another in woodwinds and strings. A concluding passage at 7 leads to a restatement of the introduction, which begins the development. The development starts with an introduction, followed by three fortissimo passages that combine materials from the first theme group and a similarly long passage, starting at 15, that does the same with the second theme group. This leads to a climax at 18, after which the recapitulation begins at 19. The recapitulation reflects the process of combining materials begun in the development, alternating forte and piano passages. This leads to a climax at 25 and the coda at 26, which uses the same materials as the introduction and ends on the combined chords from the beginning.
Throughout the piece there are tempo and time signature changes that maintain strict proportions of 3 to 4 to 5. When the time signature is 3/4, the tempo is 48; when it is 4/4, the tempo is 60, and when 5/4 it is 75. This maintains the principle of dividing a measure of four seconds into 3, 4 and 5 beats, which reflects the overall structure of passages based on trichords, tetrachords, and pentachords.
The piece is based on a series of pentachordal arrays that exclude the collection 0235 (0), or the notes C-D-E-flat-F, a group of trichordal arrays that are imbedded in these arrays, and another group of tetrachordal arrays that use the same interconnecting arrays. These arrays are the only ones that possess these properties.
Chamber Concerto No. 2 is similar to my first Chamber Concerto in that it is in two movements, the first fast and the second slow and both being based the same materials. The woodwind quartet and string quintet are treated mostly as separate groups, with the piano complementing each of them.
The first movement begins with a full passage, which is dissembled into several interlocking smaller sections, leading to a large climax. That passage is again dissembled into smaller sections, which in turn lead to some excursions away from where it began but then back into the same milieu, leading to an even bigger climax.
The second movement is in four large sections, each based on a similar and unique group of hexachordal arrays. These are made up of several smaller units, each of which is presented in a separate section and combined to form the whole. The orchestration reflects this organization, with passages for strings alternating with passages for woodwinds and the piano present in each. When combined, the whole ensemble plays. The ending is a large climax summarizing the entire process.
The first movement was sketched while I was visiting Singapore in the Fall of 2003, but I couldn’t write a note of it until returning home. The second movement was written in April of 2004.
Chamber Concerto No. 3 is the third in a series of pieces I have written for chamber orchestra ensembles, each in two movements. In this work there are five strings and five winds, and they are often pitted against one another, with the piano contributing to both parts. The first movement is created from two contrasting themes, the first stated at the beginning and the second in a soft passage involving just piano and winds. These are extended and combined extensively in the middle sections. There is a recapitulation that brings back these materials in transformed ways, leading to a climax at the end. The second movement contains two contrasting lines throughout, the winds answering the strings. Throughout much of the movement, notes fade in and out and are kept sustaining while the other groups come in. This also leads to a climax at the ending, where only the high notes that have been the genesis of the work are left sustaining.
My Sextet is based on a special set of related pentachords which can be subdivided into several similarly related sets of trichords and tetrachords. The entire work weaves through these different relationships. The chords alternate compressed harmonies, containing major and minor seconds, to expanded ones, including major seconds and perfect fourths.
It begins with quietly, with all voices stated in a single octave, gradually expanding outwards to fill five octaves, followed by a fast passage that moves through the nexus of these trichord-pentachord relationships, and then a transition. The slow and quiet middle section brings the trichords into their own, followed by a passage where they are combined with new notes to form tetrachords. The faster tempo returns, and it builds to a climax, where both trichords and pentachords are stated simultaneously. The concluding passage is like the reverse of the opening, where the voices are compressed from a full statement in five octaves to the ending, where they are all in a single octave.
The rhythms are extremely fluid, without necessarily having a strong sense of pulse. There are three proportional tempos of quarter note equaling 36, 48 and 60 beats. These tempos divide the duration of five seconds into three, four and five beats respectively, and they are related to the harmonies on which their respective passages are built.
My Trio is my first attempt to write a piece of large scope for a small number of instruments. The three short movements each consist of a connected series of passages that develop in consistent ways and lead to the next section. The only break in the sequence is at the beginning of the second movement, in which a slow passage introduces new material. Each successive passage contains elements of the previous passage, but also usually introduces new material. Each of the instruments plays both primary and supporting material. While mainly a melodic instrument, sometimes the cello plays double stops.
Breathless was written in July 2016 as a challenge to flutists and clarinettists. While I have written many solo works, when I began this work I had not written for two solo instruments in 50 years. I tried to imagine every kind of relationship that can exist between two instruments: where one takes the lead and the other follows, where both play evenly, where one instrument is entirely within one octave while the other plays in two, where both play in two, and so forth. The piece requires that both instruments play expressively over the entire range of notes that they can play, including a high C7 and C#7 on the flute. (If the player cannot play these, they can be played an octave down.) The title refers, in jest, to the fact that, except for one passage for solo clarinet, both instruments are playing continuously.
This short work was contrasts the natural properties of the clarinet and cello against each other. The two instruments must be played expressively over their entire range. The piece consists of several short sections which feature different interactions of the instruments. The title refers to the agitated and edgy character of some of the music, as well as the fact that neither instrument has a rest for the entire duration of the piece. The work was composed in September 2017.
My first composition for the keyboard was such a disaster, from the standpoint of playability, that I refrained from attempting another such undertaking for over 20 years. This was especially regretful, since my son Jonathan is a fine pianist. Finally, as I worked out how to avoid unreasonable stretches and confine the music to the range of two hands, I have written this work, which is dedicated to my son.
The piece is in three parts and ten sections. It begins with a full statement, covering five octaves, and then proceeds to an agitated passage that spans the entire seven-octave range of the piano, although in selected portions at a time, followed by a transition. The middle part is slow and lyrical, gradually intensifying, until a transition back to the texture of the opening. The third part begins with a transformed recapitulation, including both of the first two sections, to the conclusion. The ending covers a six octave range, but only three at a time, and it is arranged so that no single hand spans more than an octave and a third. Nevertheless, it is still probably extremely difficult to play.
Chimera is my first work for a solo cello, which, to me, is one of the most beautiful instruments. I rely on the cello’s ability not just to play melodies, but also double stops, which are used throughout. The piece alternates between slow and fast sections. The slow parts use lots of double stops and are generally in one octave, while the fast sections contain both leaps and double stops connecting notes in different octaves. The double stops in the slow passages use similar intervals, usually thirds, either major or minor thirds dominating, except at the ending, where minor seconds and perfect fourths are used. A chimera, in Greek mythology, is a mythical fire-breathing monster, but nowadays the term is used mainly in the figurative sense to describe an impossible or foolish fantasy. The piece was written in the summer of 2011.
Scene for Solo Clarinet is my second work for a solo melodic instrument. It consists of several short contrasting passages, often in different tempos. Each numbered section is a single unit, usually ending with a luftpause before the next. Sometimes the tempos do not change, but the amount of activity in the measure changes. I imagine the work to depict a journey through a landscape which begins rather calm and simple but extends into more and more complicated domains. After reaching a hectic climax, the work returns to a simpler context, although the effects of the previous agitation remain and gradually die down, until it finally returns to the texture of the beginning. I take advantage of the clarinet’s expressive abilities to shape phrases, to play at different dynamic levels and to bring out individual notes through swells. The piece was written during my stay in Málaga in the summer of 2012.
When you write out a succession of related chords, they are structured in two dimensions, as chords and voices, in what I usually refer to as an array. There are a remarkably small number of arrays where both dimensions can be structured with the same relationships between both the chords and the voices. These works are based on sets of tetrachords, or four-note chords, that have these properties. While, according to my theories, there are five families of tetrachords, only four of them can be structured in this way, and each of the etudes is based on a different family of tetrachords. The first and last movements use all seven octaves of the piano, while the second and third use only the more compact range of five octaves. Some passages in the first movement are written on three staves in order to clarify the voices. One of my main concerns in this work has been to make the music playable with a normal hand span for the pianist, and in this I believe that this is successful, although in some passages the fingering may be awkward.
These works continue the line of development that I began with my Tetrachordal Etudes, only the pieces are based on pentachords (five-note chords) rather than tetrachords. All of them are related in a way that gives special prominence to the notes B, F#, G# and A, and each individual piece is based on a single group of pentachords except for the third piece, which uses two. These are etudes, or studies, which develop numerous ways of presenting the same basic materials over and over. While the notes generally lie within the span of the hands, there are a few spots where the pianist has to keep melodic lines sustained in widely spaced octaves, requiring the use of the pedal. Some passages are written out on three staves to clarify these melodies. Lines run throughout the pieces in all octaves, but the harmonies are mainly pentachords. The pieces were composed in the summer of 2013.
These works continue the line of development that I began with my Tetrachordal Etudes and Pentachordal Etudes, only the pieces are based on trichords (three-note chords). There are only two sets of these, so there are two movements for each set, one of which uses only eight notes while the other uses all twelve. These are simpler than the other works, since trichords and their combinations have fewer notes than tetrachords and pentachprds. Several passages use all seven octaves of the piano and are thus separated into three staves. Melodic lines run throughout the pieces in all octaves, but the harmonies are mainly just trichords. The pieces were composed in the Fall of 2013 and Summer of 2014.
This work is a fantasy in the sense of Chopin’s fantasies, which consist of several sections in relatively free form. There are three broad sections with an extended transition between the second and third. The opening, marked “emphatically,” consists of short passages in extreme contrasting registers and dynamics that builds a mosaic which completes its structure by the ending, which comes to rest quietly in the middle register. The middle section is lyrical and expressive, which builds chords that sustain while melodies are played above and below. Sustaining chords are continued in the transition, while melodies anticipate the final section, and they are continued to the ending. In the first part of the last section, the chords move more slowly than the melodies, and the emphatic texture and tempo of the opening returns at the end, where the chords and accompanying melodies both sustain through the measures. The piece was written in the fall of 2016 and spring of 2017.
Prism is my first piece for full orchestra. It was originally written in 1976, and then revised slightly and orchestrated a bit differently in 1993, when I also did a computer-printed version of the score from the original paper manuscript. At that time I also went back and tried to figure out how I had originally constructed the composition, in the course of which I found several mistakes in the manuscript.
The piece is based on a bunch of different related materials, including arrays based on trichords, tetrachords and pentachords. There is an introduction and corresponding coda with both present different musical strands moving against each other at different tempos. The second and next-to-last sections are long, extended developments that build up to a big climax. The second section has the character of a scherzo, which was the original title of the piece, and the next-to-last, marked pesante, is more like a dance. The extended middle of the piece is a series of different sections that go through various passages in different ways. There is a 3-against-4 passage marked “Webernesque” because of its sparse texture of short woodwind tones against pizzicato strings. There are a series of short passages that alternate between 12/8 and 4/4 time; these finally culminate in a climax that leads to the “pesante”. While each seems to follow logically from the previous, the character and texture of these passages are quite different.
Symphony No. 2 was composed in 1992. It is a five-movement work with a palindromic relationship between the movements as well as the sections within the movements. The first and last movements are fast, the second and fourth slow, and the middle movement, marked "Scherzando", is moderately fast. While a palindrome is, in some sense, easy to follow, it is a challenging and interesting concept to work with, since it requires that passages that are initially heard as the opening statement of a musical idea later be heard as a conclusion or summing up, or for some other function, and vice versa. For this reason, there are some sections that are grouped together, so that the palindrome applies, for example, to a group of three sections rather than to each section individually.
The piece employs my own method of composing based upon ideas that depart markedly from the other two major methods used in the twentieth century, namely tonality and 12-tone serialism. In 12-tone music, a complex and frequently jarring musical surface is often underlaid by a simplistic background structure. Composers have resorted to such ridiculous constructions as simultaneous multiple dynamic levels and rhythms that are as impossible to play as they are to hear. Early 12-tone music employed a conscious avoidance of tonal references (such as triads or seventh chords), and to this day composers go to great lengths to avoid octaves. Since the total chromatic is always stated, differentiations are made mainly by the ordering of elements, and the entire system is governed by a logic of permutations rather than combinations that generate new elements.
In tonal music, compositions are based on simple harmonic structures (major and minor triads), and dissonances always resolve into consonances. While the dissonances are often the most interesting and important aspects of tonal compositions, for the most part tonality has not evolved to the idea of basing compositions on more complex structures than triads, and some composers who have attempted to do so have created logical inconsistencies that create impossible dilemmas.
My compositions are based on small collections of notes (3, 4, 5 or 6) related by only their intervallic structure. These are combined into groups called arrays, which possess various structural, common tone, and ordering properties that allow events to be structured in several dimensions at once. While the music may be very complex at times, the basic elements (mainly trichords, tetrachords and pentachords) are easily to perceive and to understand. This method also forces a consideration of pitch duplications not present in 12-tone music, since groups of notes combined with other groups generate new notes sometimes duplicating tones already present. On the surface of the music, this process results explicitly in pitch repetitions and octave duplications that parallel the larger structures.
The goal is a musical texture in which each note is simultaneously related to every other event in its context in several different ways. The four relationships that make up all the arrays used in the piece are the multiplicative operations of identity (M1), inversion (M11), cycle-of-fifths equivalence (M7) and inversion of cycle-of-fifths equivalence (M5), which are the only single-interval cycles within the equal_tempered scale that are capable of generating the total chromatic. Passages in the piece are based on sets of trichords, tetrachords, pentachords and hexachords related by these properties. In particular, long passages in corresponding sections are related by cycle-of-fifths equivalence.
Symphony No. 2 is based upon a particular group of arrays in which the set of notes A-Bb-B-E occur in special ways. As a result, these notes should always be followed as though they were guiding the rest of the music, which in fact they do. This process is somewhat analogous to the tonal sense of key, since all the harmonies have something to do with these notes, but not of course in the tonal sense where these are "more important" than the others and serve as the resolutions of other tones.
In listening to a composition like this, the listener should make an effort to avoid hearing events as in other music, as if the piece consisted of a tonal piece with wrong notes. Central to this process is discarding the idea of the resolution of notes in a dissonant chord into a consonance. Although many events in the piece are reminiscent of tonality -- and indeed, I have often striven to use elements like triads -- the traditional concepts of tonal resolutions and the sense of key are not present. Dissonances (indeed, any structures) are simply stated, and the intervallic relationship between the constituent notes should be perceived as the essence of the total structure. Rather than looking for resolutions, the listener should direct his or her attention to looking for similarities between the present sounds and others in the immediate and longer-range context, keeping in mind that, in inversions, minor seconds are exchanged with major sevenths, and in cycle-of-fifths equivalence, with perfect fifths.
The piece is scored for chamber orchestra consisting of one flute, oboe, clarinet, bassoon, french horn, trumpet, and piano, and strings in sections (at least two on each part). In particular, the cello and string bass parts are different, and the string bass part can be played completely on cello if a string bass is not available.
Each of my symphonies is based on a set of related materials for which I can see numerous possibilities for working things out, thus leading to several different movements. Symphony No. 3 is based on a series of trichords, tetrachords and pentachords that all relate to the diminished seventh chord. The instrumentation, consisting of four woodwinds, four brasses, piano and strings, allows different materials to be presented in distinct instrumental groups, and when combined, to keep the threads clear. The piano, which has several solos in the quieter moments, combines with all the other groups.
In the first movement, there are two principal ideas, the first based on pentachords and presented in a faster tempo, and the second on tetrachords and trichords. After a slow introduction, these materials are presented in a series of evolving sections, followed by an extensive development that combines different parts of each and then the two together, reaching an intense climax. This is followed by a transformed recapitulation and conclusion.
The second movement presents two overlapping strands of music, identified by orchestration. The first strand begins in the winds and brass, and the second in the strings. Each strand proceeds by adding new elements to the preceding passage, or by extracting elements from it. Thus, there are many lines that are repeated, each time in a different context. The movement builds to a climax and then recedes.
The third movement is based entirely on tetrachords. The first two sections run through the basic forms used, and the middle section runs through basically all different combinations and permutations of these materials. After a transformed recapitulation, the work ends with a brief coda.
Symphony No. 4 is based largely on the same materials as my first symphony, yet it sounds completely different, at least on the surface. There are three movements, each of them quite different. The chamber orchestra includes four woodwinds, four brasses, piano and strings, and orchestration is used to articulate contrasting voices.
The first movement is similar to a classical sonata form in that includes an introduction, exposition, development, recapitulation and coda, and is based on two contrasting ideas which are combined and extended to form the central development section of the movement. The introduction presents these materials simultaneously, with the brass instruments playing a slower passage connected to the second part while the strings and woodwinds play material from the first. The first section (rehearsal numbers 2 to 10) unfolds the materials of the first part in groups of three sections, each with a softer passage in the middle. The tempo and meter change after the first group, and the third group is a climax for this part. The second section (from 11 to 15) is quiet and is presented only in the woodwinds and strings, except for two notes on the French horn. The next two brief sections (16 and 17) are a transition to the development, the second one being a huge climax. The development is in two parts, based on the materials from the first two sections, although now each passage is combined with other materials. The first section (18-26) builds substantial counterpoint between the woodwinds, brasses and strings, each group sometimes presenting separate lines, with the piano combining with each of them in different spots. The second section (27-30) returns to the quietude and slow tempo of the exposition, with a prominent piano solo. It builds and leads logically to the recapitulation (31), which is a climax in the same tempo but with the brasses moving more slowly than the other instruments, as in the beginning. The recapitulation (32-40 and 41-44) parallels the exposition, but each section is a transformation of the corresponding passage in the beginning. The coda (45) is in a slower tempo and is a kind of summary and conclusion of the movement.
The second movement, inspired by a musical passage I heard over 30 years ago, presents each instrument playing exactly the same thing, combining in different ways to create different harmonies. I call this idea a “freeze,” and I first used it in an electronic composition I wrote in 1972. In the first and last of the movement’s three parts (46-52 and 58-65), each instrument plays just one note, in the same rhythm each time it occurs, although the rhythms of the two sections are different. In the middle part (53-57), which is the only one that contains a real climax, some instruments play two notes, including the notes that they play in the other parts. The “freeze” is not completely strict, because some of the brass instruments as well as the two violins exchange their notes, but the other do not.
The last movement is a kind of rondo in an A-B-A-C-A-D-A-Coda form. The only way that this differs from the traditional rondo is that each of the “A” sections (66-67, 71-72, 81-82, 91-92) is a new variation, maintaining the same rhythm but interchanging the notes. Each of the intervening sections (68-70, 73-80, 83-90), is an excursion that leads through several passages and sometimes through dense combinations, only to wend its way back to the beginning. The ending (93) is a big climax, followed by a quiet “residue.” Throughout the movement most of the winds instruments play long solo notes, while the strings provide the rhythmic and harmonic motion.
Expansions 2 is a piece that states a simple underlying musical structure, where each note is then “expanded” by a sequence of other notes, both harmonically and melodically. The expansions are stated in rhythms that extend the original note and span as much as three measures. In most of the piece, the original structure is stated in the woodwinds and the expansions in the strings. The expansions range from as little as three notes to as many as nine. The textures sometimes become very dense, with multiple overlappings and extensions. Once stated, notes longer than a few beats fade in a diminuendo, and much of the surface of the piece is dominated by notes fading out as others keep entering. The piece alternates sections of various densities and dynamics, and, in the middle, the expansions become extended melodies. A climax occurs when the expansions include both harmonies and melodies, after which the work returns to the opening texture. The piece was written in the fall of 2017.
Expansions 2 is the second work developing an idea which I originally created in a piece of electronic music (Expansions). In other compositions that I have written, each tone includes several different components that fade in and out over the course of the duration. However, in most of those cases, the components were inharmonic. Expansions was the first work I attempted where the components were themselves tempered pitches, and thus capable of being played by separate instruments. That work has not been orchestrated.
Clusters for Wind Ensemble was inspired by my electronic piece of the same name. In that work, the main idea was creating the spectrum of the sounds from the harmony of the passages in which the sounds occur. In this work, groups of instruments play clusters in a more direct manner over a slowly-shifting passage that is always stated in the brass instruments. While the brass provides the foundation, four-note clusters above the basic tone are stated in various choirs of similar instruments – flutes, clarinets, saxophones, and various other combinations of similar instruments. The piano plays these clusters as well in the louder passages. Percussion, including marimba and xylophone, play scalar passages, and chimes enter at the climactic moment. Except for two loud passages, the work is mainly soft and transparent, with choirs of instruments fading in and out asynchronously against one another. The work was written in 2010.